Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245063 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35150 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1382 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,25)(20,26)(21,27)(22,35)(23,30)(24,31)(28,33)(29,36)(32,39)(34,38)(37,40);(9,11,14,21)(10,15,18,27)(12,19,23,33)(13,20,24,34)(16,28,30,25)(17,26,31,38)(22,32,36,40)(29,39,35,37);(2,5)(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,29)(18,30)(20,32)(21,33)(24,36)(26,37)(27,28)(31,35)(34,40)(38,39);(9,13,14,24)(10,17,18,31)(11,20,21,34)(12,22,23,36)(15,26,27,38)(16,29,30,35)(19,32,33,40)(25,37,28,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152265);